How to Add and Subtract
Positive and Negative Numbers

Numbers Can be Positive or Negative:

number line -10 to +10
Negative Numbers (−) Positive Numbers (+)
(This is the Number Line, read about Using The Number Line)

minus plus
"−" is the negative sign. "+" is the positive sign

No Sign Means Positive

If a number has no sign it usually means that it is a positive number.

Example: 5 is really +5

Adding Positive Numbers

Adding positive numbers is just simple addition.

Example: 2 + 3 = 5

is really saying

"Positive 2 plus Positive 3 equals Positive 5"

 

You could write it as (+2) + (+3) = (+5)

Subtracting Positive Numbers

Subtracting positive numbers is just simple subtraction.

Example: 6 − 3 = 3

is really saying

"Positive 6 minus Positive 3 equals Positive 3"

 

You could write it as (+6) − (+3) = (+3)

Balloons and Weights

add subtract general  

This basket has balloons and weights tied to it:

  • The balloons pull up (positive)
  • And the weights drag down (negative)

 

Here is what adding and subtracting positive numbers looks like:

add subtract ++  goes up  

You can add balloons (you are adding positive value)

the basket gets pulled upwards (positive)

 

add subtract -+  goes down  

You can take away balloons (you are subtracting positive value)

the basket gets pulled downwards (negative)

 

Now let's see what adding and subtracting negative numbers looks like:

add subtract +- goes down  

You can add weights (you are adding negative values)

the basket gets pulled downwards (negative)

 

add subtract -- goes up  

And you can take away weights (you are subtracting negative values)

the basket gets pulled upwards (positive)

 

That last one was interesting ... subtracting a negative made the basket go up

minus minus gives plus

Subtracting a Negative is the same as Adding

Example: What is 6 − (−3) ?

6−(−3) = 6 + 3 = 9

Example: What is 14 − (−4) ?

14−(−4) = 14 + 4 = 18

 

We also found that taking away balloons (subtracting positives) or adding weights (adding negatives) both made the basket go down.

And Positive and Negative Together ...

plus minus gives minus Subtracting a Positive
or
Adding a Negative

is
Subtraction

Example: What is 6 − (+3) ?

6−(+3) = 6 3 = 3

Example: What is 5 + (−7) ?

5+(−7) = 5 7 = −2

 

The Rules:

It can all be put into two rules:

  Rule   Example
plus Two like signs become a positive sign +(+) 3+(+2) = 3 + 2 = 5
−(−) 6−(−3) = 6 + 3 = 9
       
minus Two unlike signs become a negative sign +(−) 7+(−2) = 7 2 = 5
−(+) 8−(+2) = 8 2 = 6
       

They are "like signs" when they are like each other (in other words: the same).

 

So, all you have to remember is:

Two like signs become a positive sign

Two unlike signs become a negative sign


Example: What is 5+(−2) ?

+(−) are unlike signs (they are not the same), so they become a negative sign.

5+(−2) = 5 2 = 3

Example: What is 25−(−4) ?

−(−) are like signs, so they become a positive sign.

25−(−4) = 25+4 = 29

Example: What is −6+(+3) ?

+(+) are like signs, so they become a positive sign.

−6+(+3) = −6 + 3 = -3

Start at −6 on the number line, move forward 3, and you end up at −3

Now Play With It!

pat athlete   Try playing Casey Runner, you need to know the rules of positive and negative to succeed!

 

A Common Sense Explanation

And there is a "common sense" explanation:

dog brocolli

If I say "Eat!" I am encouraging you to eat (positive)

But if I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

So, two negatives make a positive, and if that satisfies you, then you are done!

 

Another Common Sense Explanation

A friend is +, an enemy is

+ + ⇒ +     a friend of a friend is my friend
+ − ⇒ −     a friend of an enemy is my enemy
− + ⇒ −     an enemy of a friend is my enemy
− − ⇒ +     an enemy of an enemy is my friend

A Bank Example

Example: You have $80 in your account right now.

Last year the bank subtracted $10 by mistake, and they want to fix it.

So the bank must take away a negative $10:

$80−(−$10) = $80 + $10 = $90

So you get $10 more in your account.

Another Example You Might Like

Ally's Points

alexa

Ally can be naughty or nice. So Ally's parents have said

"If you are nice we will add 3 points (+3).
If you are naughty, we take away 3 points (−3).
When you reach 30 Points you get a toy."

Ally starts the day with 9 Points:

 
Ally's Mom discovers spilt milk: 9−3 = 6
 
Then Dad confesses he spilt the milk and writes "undo". Mom calculates: 6−(−3) = 6+3 = 9
alexas points

So if you subtract a negative, you gain points
(i.e. the same as adding points).

minus minus gives plus

So Subtracting a Negative is the same as Adding


 

 
A few days later. Ally has 12 points.  
Mom adds 3 points because Ally's room is clean. 12+3 = 15
Dad says "I cleaned that room" and writes "undo" on the chart. Mom calculates: 15 − (+3) = 12
Dad sees Ally brushing the dog. Writes "+3" on the chart. Mom calculates: 12 + (+3) = 15
Ally throws a stone against the window. Dad writes "−3" on the chart. Mom calculates: 15 + (−3) = 12
alexas points2

See: both "15 − (+3)" and "15 + (−3)" result in 12.

So:

It doesn't matter if you subtract positive points
or add negative points,
you still end up losing points.

plus minus gives minus So, Subtracting a Positive
or
Adding a Negative

is
Subtraction

 

Try These Exercises ...

Now try This Worksheet, and see how you go.

And also try these questions: